Conjunctive Types and SKInT

1 Introduction The-calculus and its typed versions are important tools for studying most fundamental computation and deduction paradigms. However, the non-trivial nature of substitution, as used in the deenition of-reduction notably, has spurred the design of various rst-order languages representing-terms,-reduction and-conversion, where computation is simple, rst-order rewriting, and substitution becomes an easy notion again. Let us cite Curry's combinators 7], Curien's categorical combinators 5], the myriad of so-called-calculi with explicit substitutions , among which 1], * 12], 17], 19], etc. Unfortunately, each one of these calculi has defects: Curry's combinators do not model-conversion fully, categorical combinators, and * do not normalize strongly in the typed case 18], is not connuent in the presence of free variables (a.k.a. meta-variables), models-conversion but not-reduction, etc. SKInT 9] is a rst-order language and rewrite system that does not have these defects. In particular, SKInT models reduction in the-calculus, in the sense that there is a mapping L from-terms to SKInT such that whenever u rewrites to v in the-calculus, then L (u) rewrites to L (v) in SKInT. SKInT is connuent even on open terms, i.e. terms with meta-variables; L deenes a conservative embedding of the-calculus inside SKInT, in that u and v are-convertible if and only if L (u) and L (v) are convertible in SKInT; reduction in SKInT standardizes; L preserves weak normalization, i.e., if u is a weakly normalizing-term, then L (u) is a weakly normalizing SKInT-term. SKInT also enjoys a simple type discipline corresponding to that of the-calculus, that is, if u is a-term of type , then L (u) is a SKInT-term of some type L () easily computed from ; reduction in SKInT obeys subject reduction, and every simply-typed SKInT-term normalizes strongly 9]. The aim of this paper is to extend our result on preservation of weak normal-ization to show that SKInT, like and , also preserves strong normalization, and also solvability; this is done by showing that, just like in the-calculus, strongly normalizing, weakly normalizing and solvable terms are characterized as terms that are typable in various conjunctive type disciplines 20, 4]. The plan of the paper is as follows: we introduce the required notions and notations in Section 2, then we attack our goal in Section 3. We show that any reasonable translation from the-calculus to SKInT preserves solvability, weak ? This work has been done in the context of Dyade (Bull/Inria R&D joint venture).